Each class of objects has an abundance of objects that are Kolmogorov random relative to the class. The incompressibility method has been successfully applied to solve open problems and simplify existing proofs. The method rests on a simple fact: a Kolmogorov random string cannot be compressed. These are the incompressible objects. Although individual objects cannot be proved to be incompressible in any given finite axiom system, a simple counting argument shows that almost all objects are incompressible.
In a typical proof using the incompressibility method, one first chooses a Kolmogorov random object from the class under discussion. This object is incompressible.
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Then one proves that the desired property holds for this object. The argument invariably says that if the property does not hold, then the object can be compressed.
This yields the required contradiction. Because we are dealing with only one fixed object, the resulting proofs tend to be simple and natural. They are natural in that they supply rigorous analogues for our intuitive reasoning. In many cases a proof using the incompressibility method implies an average-case result since almost all strings are incompressible. The method is always a matter of using regularity in an object, or algorithm , imposed by a property under investigation and quantified in an assumption to be contradicted, to compress the object's or algorithm's description to below its minimal value.
The incompressibility method is the oldest and the most used application of algorithmic complexity ,. We first prove, following G. Can we do better?
Combinatorial Geometry and its Algorithmic Applications: The Alcala Lectures
This is slightly more complicated. The original idea is due to P. Berman, and improved by J. He constructed such a statement. Here we use the incompressibility argument to show in a very simple manner that there are, in fact, infinitely many such undecidable statements. A formal system consisting of definitions, axioms, rules of inference is consistent if no statement that can be expressed in the system can be proved to be both true and false in the system.
A formal system is sound if only true statements can be proved to be true in the system. Hence, a sound formal system is consistent. The idea below goes back to Ya. Barzdins and was popularized by G. Hence, we have a contradiction. This shows that although most strings are random, it is impossible to effectively prove them random. In a way, this explains why the incompressibility method is so successful. The incompressibility method has been very successful to analyze the difficult average case complexity of algorithms. The method simply analyzes the algorithm with respect to this single string, by showing that the string is compressible if the algorithm does not satisfy a certain running time.
This method has been used to analyze the average case running time for many well-known sorting algorithms, including the Heapsort and the Shellsort algorithm. This is the first nontrivial general lower bound for average-case Shellsort in 40 years. Traditional wisdom has it that the better a theory compresses the learning data concerning some phenomenon under investigation, the better we learn, generalize, and the better the theory predicts unknown data.
This belief is vindicated in practice but before the advent of Kolmogorov complexity has not been rigorously proved in a general setting. The material on applications of compressibility is covered in Li and Vitanyi , Chapter 5.
Applications of algorithmic information theory - Scholarpedia
Ray Solomonoff invented the notion of universal prediction using the Kolmogorov complexity based universal distribution, see the section on Algorithmic Probability in Algorithmic Information Theory. Universal prediction is related to optimal effective compression. The latter is almost always a best strategy in hypotheses identification the minimum description length MDL principle.
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While most strings are incompressible, they represent data where there is no meaningful law or regularity to learn; it is precisely the compressible strings that represent data where we can learn meaningful laws from. As perhaps the last mathematical innovation of an extraordinary scientific career, Kolmogorov in proposed to found statistical theory on finite combinatorial principles independent of probabilistic assumptions. Technically, the new statistics is expressed in terms of Kolmogorov complexity. The relation between the individual data and its explanation model is expressed by Kolmogorov's Structure function.
This entails a non-probabilistic approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal Kolmogorov complexity. The Structure function of the given data expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data. Essentially, for given data, one analysis by Vitanyi and Li tells which models obtained by the MDL principle are the right ones, the best fitting ones, while the analysis using the structure function tells how to obtain them.
In this setting, this happens with certainty, rather than with high probability as is in the classical case. Related ideas have been applied, by Vereshchagin and Vitanyi to rate-distortion theory and lossy compression of individual data.
Combinatorial Geometry with Algorithmic Applications
Cognitive psychology has a long tradition of applying formal models of simplicity and complexity. The work by E. Leeuwenberg even predates the advent of Kolmogorov complexity. Not surprisingly, this field has a large and significant literature on applications of Kolmogorov complexity, for example the circles around N. Chater and P. Some strings can be compressed but take a great amount of effort, in time or space, to do so. Applications of compressibility requiring a great effort are covered in Li and Vitanyi , Chapter 7.
In particular, she used probabilistic methods to solve several problems on Hamilton cycles in graphs and digraphs, graph decompositions and hypergraph matchings. Deryk's research interests are in extremal graph theory, random graphs, randomized algorithms, structural graph theory as well as Ramsey theory.
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His recent research has included results on Hamilton cycles and more general spanning substructures, as well as decompositions of graphs and hypergraphs. Nikolaos' research interests are mainly related to the area of random discrete structures and the analysis of random processes on graphs and their connections with theoretical computer science and average-case analysis. His most recent work has focused on the development of the theory of random graphs on the hyperbolic plane and its applications to the theory of complex networks.
He is also interested in percolation phenomena in large finite structures. Richard's research is primarily in the field of extremal graph theory. Recent results include general sufficient conditions which ensure the existence of perfect matchings and Hamilton cycles in hypergraphs, or which permit the construction of efficient algorithms to find such structures should they exist. He is also interested in graph decompositions and on a number of embedding problems in the directed graph and hypergraph setting. Allan's research interests lie in Extremal and Probabilistic Graph Theory. A typical problem in this field is to determine the necessary conditions for the existence of a fixed spanning subgraph in a graph, edge-coloured graph, orientated graph or hypergraph.
Johannes is particularly interested in Graph Minors, Connectivity and Matroids. Recently he characterised the simply connected 2-complexes embeddable in 3-space -- in a way similar to Kuratowski's characterisation of graph planarity. Richard's research interests lie mainly in Extremal and Probabilistic Graph Theory.